Grade 4 · 4.OA.A.1 · 4.OA.A.2

Times as many

One pile has 7 stickers and another has 28. Saying “21 more” is true, but how do you say that the big pile is four of the small one?

You need

What multiplication meansGrade 2

Where you are 16 in Multiplication and division

Number9

Arithmetic5

Multiplication and division16

▴ 3 above Multiplication facts4 The distributive property

Times as many

What division means Division facts ▾ Show all 16 (8 more)

Why this exists

Killa has 7 stickers. Chirag has 28. The quick way to compare them is to find the gap: 28 - 7 = 21, so Chirag has 21 more. That is true, and it is the comparison taught in Subtraction as the difference. But it leaves something out.

Look at a second pair. One sheet has 100 stickers and another has 121. The gap is 21 again. Yet the two pairs are not alike at all. A sheet of 121 is only a little bigger than a sheet of 100. A pile of 28 is four whole piles of 7. The gap cannot tell these two pairs apart.

So there is a second way to compare: count how many copies of the smaller amount it takes to make the bigger one. Chirag’s 28 is 4 copies of Killa’s 7, so Chirag has 4 times as many stickers as Killa. This kind of comparison says how big one amount is next to the other, and it is the comparison that a building “three times as tall” or a recipe “twice as much flour” uses.

Definition

“One amount is 4 times as many as another” means the first amount is made of 4 copies of the second, put together. In symbols, 28 is 4 times as many as 7 because

28 = 4 \times 7

In general, for whole numbers a, b and n: a is n times as many as b when a = n \times b. The number n says how many copies of b make a.

The same words work for any kind of amount that can be copied: “times as tall”, “times as long”, “times as heavy”, “times as far”. Both amounts must count or measure the same kind of thing in the same unit. This is a multiplicative comparison, because it compares by multiplying. A comparison by a gap (“21 more”, “21 fewer”) is an additive comparison, because it compares by adding on.

A reminder of what 4 \times 7 means, from What multiplication means: 4 equal groups of 7, which is 7 + 7 + 7 + 7.

Intuition

Part 1: times as many is copies, read both ways

Draw Killa’s stickers as a bar. Then draw Chirag’s stickers as a bar made of boxes, each box as long as Killa’s whole bar. Four boxes fill Chirag’s bar exactly.

Figure 1Chirag’s 28 stickers are four copies of Killa’s 7.

A picture like this is a tape diagram: amounts drawn as bars, cut into equal boxes. The boxes do not need a square for every sticker. What matters is that every box in the long bar is the same size as the short bar. Count the boxes and you have the times. Multiply and you have the total: 4 \times 7 = 28.

Now cut the same length a different way. The 28 can be cut into 4 boxes of 7, or into 7 boxes of 4.

Figure 2The same 28 is four sevens and seven fours.

Both bars have the same length, 28. So the one equation 28 = 4 \times 7 says two things:

28 is 4 times as many as 7 (four sevens), and
28 is 7 times as many as 4 (seven fours).

The two factors can trade places because 4 \times 7 = 7 \times 4. That is the commutative property of multiplication: turn an array of 4 rows of 7 on its side and it is 7 rows of 4, with the same number of items.

Part 2: “more” and “times as many” answer different questions

Put Killa’s bar under Chirag’s again. Line up Killa’s box with Chirag’s first box.

Figure 3One pair of bars, two true comparisons. The bracket is the gap; the count of boxes is the times.

One picture, two true comparisons:

Additive. The part of Chirag’s bar that sticks out past Killa’s is the gap: 28 - 7 = 21. Chirag has 21 more stickers than Killa. This answers “how many more?”
Multiplicative. Chirag’s bar holds 4 boxes the size of Killa’s: 28 = 4 \times 7. Chirag has 4 times as many stickers as Killa. This answers “how many times as many?”

The two answers are different numbers, 21 and 4, because they answer different questions. A story’s question tells you which one it wants. “How many more”, “how many fewer”, “how much taller” ask for the gap. “How many times as many”, “how many times as tall” ask for the number of copies.

Notice what the gap is in the picture: the three boxes after the first. So when one amount is 4 times as many as another, it is 3 copies more. The gap counts the extra copies, the times counts all of them.

A word about wording. People sometimes say “4 times more than”. That phrase is ambiguous: some readers take it to mean 4 times as many, and others take it to mean the amount plus 4 times the amount on top. This lesson says “times as many” and never “times more than”, because “times as many” has only one meaning.

Part 3: three numbers, and the question mark in any place

Every multiplicative comparison has three numbers: the smaller amount (one box), the times (how many boxes), and the bigger amount (the whole long bar). They are tied together by

\text{bigger} = \text{times} \times \text{smaller}

A story gives two of the three and asks for the third. The question mark can sit in any of three places.

Figure 4The question mark can sit in any of three places.

A: the bigger amount is missing. We know the box and how many boxes. Put the copies together: multiply. 4 \times 7 = \square, so \square = 28.
B: the smaller amount is missing. We know the whole bar and how many equal boxes it is cut into. Share the whole into that many equal parts: divide. 4 \times \square = 28, so \square = 28 \div 4 = 7.
C: the times is missing. We know the whole bar and the size of one box. Find how many boxes fit: divide. \square \times 7 = 28, so \square = 28 \div 7 = 4.

So a “times as many” story is solved by multiplying when the bigger amount is missing and by dividing when either of the other two numbers is missing. Division is the right tool because it undoes multiplication, as Multiplication and division undo each other explains: 4 \times 7 = 28 goes with both 28 \div 4 = 7 and 28 \div 7 = 4.

Results

The meaning. For whole numbers a, b and n,

a \text{ is } n \text{ times as many as } b \quad\text{means}\quad a = n \times b

Both ways. If a = n \times b, then a is n times as many as b, and a is also b times as many as n.

The three unknowns.

Missing	Equation	Solve by
the bigger amount	n \times b = \square	multiplying, n \times b
the smaller amount	n \times \square = a	dividing, a \div n
the times	\square \times b = a	dividing, a \div b

Gap and times. If the bigger amount is n times as many as the smaller, the gap is n - 1 copies of the smaller:

a - b = (n - 1) \times b

Why it is true

“a is n times as many as b” is defined as “a is n copies of b put together”, and n equal groups of b is exactly what n \times b means. So the first result is the definition written in symbols.

For “both ways”, arrange the n copies of b as an array with n rows of b. Turned on its side, the same items form b rows of n. Nothing was added or taken away, so n \times b = b \times n, and a is also b copies of n.

For the missing smaller amount, the bigger amount a is cut into n equal boxes, and one box is the smaller amount. Cutting a whole into n equal parts and asking for the size of one part is what a \div n means. For the missing times, we ask how many boxes of size b fit into a, and “how many groups of b make a” is the other meaning of a \div b. Both meanings of division are in What division means.

For the gap, line up the single box of b under the first of the n boxes. That first box has a partner; the other n - 1 boxes do not. The unmatched part is the gap, and it is n - 1 boxes of size b. \blacksquare

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, and Examples 5, 6 and 7 with Part 3.

Example 1

One equation, two sentences

(a) Write two “times as many” sentences for 54 = 6 \times 9. (b) Write an equation for “72 is 8 times as many as 9”, then give its second sentence.

Solution

(a) The equation says 54 is 6 copies of 9, so 54 is 6 times as many as 9. By the commutative property, 6 \times 9 = 9 \times 6, so 54 is also 9 copies of 6: 54 is 9 times as many as 6.

(b) “8 times as many as 9” means 8 copies of 9, which is 8 \times 9. The sentence says that equals 72:

72 = 8 \times 9

Check: 8 \times 9 = 72 is true. Reading the factors the other way, 72 is also 9 times as many as 8.

Example 2

Reading a tape diagram

Here is a tape diagram of Manaia’s stickers and Killa’s stickers. Write a “times as many” sentence that it shows, write an equation, and find how many stickers Manaia has.

Figure 5The tape diagram for Example 2.

Solution

Killa’s bar is one box of 9. Manaia’s bar is made of boxes the same size as Killa’s bar, and there are 5 of them. So Manaia has 5 times as many stickers as Killa. The bracket over Manaia’s whole bar asks for her total, which is 5 copies of 9:

5 \times 9 = 45

Manaia has 45 stickers. Check by adding the boxes: 9 + 9 + 9 + 9 + 9 = 45. The same equation also says 45 is 9 times as many as 5, though the picture shows the first reading.

Example 3

Two comparisons of one pair

Chirag has 40 stickers and Killa has 8. (a) How many more stickers does Chirag have than Killa? (b) How many times as many stickers does Chirag have as Killa?

Solution

(a) “How many more” asks for the gap. Take the smaller amount from the bigger:

40 - 8 = 32

Chirag has 32 more stickers.

(b) “How many times as many” asks how many copies of 8 make 40. The equation is \square \times 8 = 40. Since 5 \times 8 = 40, the box is 5; that is the same as 40 \div 8 = 5. Chirag has 5 times as many stickers as Killa.

Both answers describe the same two piles. They differ because the questions differ. Check that they agree with each other: 5 copies of 8 is 1 copy to match Killa and 4 extra, and 4 \times 8 = 32, the gap from part (a).

Example 4

“More” or “times”?

Killa’s sticker sheet has 9 stickers. (a) Manaia’s sheet has 3 more stickers than Killa’s. (b) Chirag’s sheet has 3 times as many stickers as Killa’s. How many stickers are on each sheet?

Solution

The two stories use the same numbers, 9 and 3. The words between them decide what to do.

(a) “3 more than 9” is an additive comparison. Manaia’s sheet is Killa’s 9 and 3 extra:

9 + 3 = 12

(b) “3 times as many as 9” is a multiplicative comparison. Chirag’s sheet is 3 copies of Killa’s 9:

3 \times 9 = 27

Manaia’s sheet has 12 stickers and Chirag’s has 27. “More” adds on a fixed number; “times as many” makes copies. With a tape diagram, (a) is Killa’s bar with a short piece of 3 on the end, and (b) is three bars as long as Killa’s.

Example 5

Finding the bigger amount

The Pell Building is 50 meters tall. The Juniper Tower is 6 times as tall as the Pell Building. How tall is the Juniper Tower?

Solution

The Pell Building is the smaller amount, one box. The Juniper Tower is 6 boxes of that size, and its height is missing: the question mark sits over the whole long bar.

Figure 6The Juniper Tower is six copies of the Pell Building’s height.

The bigger amount is missing, so we multiply:

6 \times 50 = \square

Think of 50 as 5 tens. Six copies of 5 tens is 6 \times 5 = 30 tens, and 30 tens is 300. So \square = 300. Working in tens like this is taken further in Multiplying by tens and hundreds.

The Juniper Tower is 300 meters tall. Check: 300 \div 6 = 50, the Pell Building’s height.

Example 6

Finding the smaller amount

The Lark Tower is 360 meters tall. It is 4 times as tall as Birch Hall. How tall is Birch Hall?

Solution

Here the height we know is the bigger one. The Lark Tower is 4 copies of Birch Hall’s height, and we know the Lark Tower. So the long bar is known, 360, and is cut into 4 equal boxes; one box, Birch Hall, is missing.

Figure 7The whole is known; one equal box is missing.

The equation is

4 \times \square = 360

and since the smaller amount is missing, we divide: \square = 360 \div 4. Think of 360 as 36 tens. Sharing 36 tens into 4 equal parts gives 9 tens in each part, because 4 \times 9 = 36. Nine tens is 90.

Birch Hall is 90 meters tall. Check: 4 \times 90 = 360, so the Lark Tower is indeed 4 times as tall. Multiplying here would have given 4 \times 360 = 1{,}440 meters for the shorter building, which the story rules out.

Example 7

How many times, and how much taller

The Cobalt Tower is 320 meters tall and the Ash Building is 40 meters tall. (a) How many times as tall as the Ash Building is the Cobalt Tower? (b) How many meters taller is it?

Solution

(a) Both heights are known; the number of copies is missing. The equation is

\square \times 40 = 320

so we divide, \square = 320 \div 40. Count in tens: 320 is 32 tens and 40 is 4 tens. The question becomes how many fours make 32, and 8 \times 4 = 32. So 8 \times 40 = 320, and \square = 8. The Cobalt Tower is 8 times as tall as the Ash Building.

(b) “How many meters taller” asks for the gap:

320 - 40 = 280

The Cobalt Tower is 280 meters taller. Check with the boxes: 8 boxes of 40 meters, one of them matched by the Ash Building, leaves 8 - 1 = 7 extra boxes, and 7 \times 40 = 280. Both answers come from the same tape diagram.

Boundaries

Whole numbers only, here. In this lesson the times is a whole number of copies. “Half as tall” or “2 and a half times as many” need fractions, which come later.
“How many times” needs an exact fit. 30 is not a whole number of times as many as 8: 3 \times 8 = 24 is too small and 4 \times 8 = 32 is too big. With whole numbers, a is a whole number of times as many as b only when b fits into a exactly.
The smaller amount cannot be zero in a “how many times” question. Any number of copies of 0 is still 0, so no number of copies of 0 makes 28, and every number of copies of 0 makes 0. The question has no single answer.
One time as many is the same amount. 1 \times 7 = 7, so “1 time as many” means equal, and the gap is 0.
Same kind of thing, same unit. 28 stickers is not 4 times as many as 7 sheets of stickers. Heights must both be in meters, or both in feet, before one is called so many times the other.
Not “times more than”. That phrase has two readings, as Part 2 explains. Say “times as many” or “times as tall”.

Where this goes next

“Times as many” is the first use of multiplication to compare rather than to count groups, and it turns up everywhere in the rest of the course. The towers in Part 3 already needed products like 6 \times 50; the next lessons, beginning with Multiplying and dividing by ten, one hundred and one thousand and Multiplying by tens and hundreds, make products of bigger numbers easy to find, and Equal group problems and Problems using several operations put comparisons into longer stories. Expressions such as “three times as large as 18 + 7” in Writing and reading numerical expressions from words read a “times as many” sentence straight into symbols.

Comparing by copies is also the seed of the ratio: “for every 1 sticker Killa has, Chirag has 4” is the same picture, and the bars become the tape diagrams for ratios. Later still, multiplying by a fraction turns “times as many” into “times as large” for numbers smaller than one, which is scaling.

Exercises

Exercises 1 to 8 go with Part 1, 9 to 15 with Part 2, and 16 to 22 with Part 3. The reasoning group draws on all three.

Part 1: times as many, read both ways

Write two “times as many” sentences for 24 = 3 \times 8.
Write two “times as many” sentences for 42 = 6 \times 7.
Write an equation for “64 is 8 times as many as 8”. Why does this one have only one sentence?
Write an equation for “50 is 5 times as many as 10”, then give its second sentence.
Chirag has 6 stickers. Manaia has 5 times as many stickers as Chirag. Draw a tape diagram and find how many stickers Manaia has.
A tape diagram shows one bar that is a single box marked 4, and a second bar of 9 boxes, each marked 4. Write the “times as many” sentence it shows and an equation.
Fill in the blanks: 63 is \square times as many as 9, and 63 is \square times as many as 7.
Which are true? (a) 18 is 3 times as many as 6. (b) 18 is 6 times as many as 3. (c) 18 is 2 times as many as 8.

Part 2: more, or times as many

Killa has 32 stickers and Chirag has 8. How many more stickers does Killa have? How many times as many?
Manaia has 5 stickers. Chirag has 4 more stickers than Manaia. Killa has 4 times as many stickers as Manaia. How many stickers do Chirag and Killa each have?
Fill in the blanks: 48 is \square more than 6, and 48 is \square times as many as 6.
A small sticker sheet has 6 stickers. (a) A big sheet has 3 times as many. (b) A middle sheet has 3 more. How many stickers are on each sheet?
Chirag’s pile is 3 times as many stickers as Killa’s pile, and it is also 12 more. How many stickers are in each pile? (Hint: draw the tape diagram and find the extra boxes.)
Manaia says, “24 is 3 times as many as 8, so 24 is 3 more than 8.” Is she right? Find how many more 24 is than 8.
Killa has 35 stickers and Manaia has 7. Answer both: how many more, and how many times as many?

Part 3: which number is missing

The Pell Building is 30 meters tall. The Juniper Tower is 8 times as tall. How tall is the Juniper Tower?
Manaia has 7 stickers. Killa has 9 times as many stickers as Manaia. How many stickers does Killa have?
The Maple Building is 200 meters tall. That is 5 times as tall as the clock tower beside it. How tall is the clock tower?
Chirag has 56 stickers. That is 7 times as many as Killa has. How many stickers does Killa have?
The Summit Tower is 420 meters tall and the Fern Building is 70 meters tall. How many times as tall as the Fern Building is the Summit Tower?
Manaia’s sticker book holds 48 stickers, 6 times as many as one sheet holds. Write an equation with a \square and find how many stickers one sheet holds.
Chirag has 7 stickers. Killa has 3 times as many as Chirag, and Manaia has 5 more than Killa. How many stickers does Manaia have?

Reasoning

Use an array to explain why 30 = 5 \times 6 means both “30 is 5 times as many as 6” and “30 is 6 times as many as 5”.
Is 25 a whole number of times as many as 4? Explain.
Explain why, when one amount is 4 times as many as another, it is 3 copies of the other more. Use this to find how many more 4 times as many as 9 is than 9.

Answers

24 is 3 times as many as 8, and 24 is 8 times as many as 3. Both come from 3 \times 8 = 8 \times 3 = 24.
42 is 6 times as many as 7, and 42 is 7 times as many as 6.
64 = 8 \times 8. Both factors are 8, so trading them gives back the same sentence.
50 = 5 \times 10. Second sentence: 50 is 10 times as many as 5.
30 stickers. The tape diagram is Chirag’s one box of 6 and Manaia’s bar of 5 boxes of 6, so 5 \times 6 = 30.
The second bar is 9 times as many as 4: 36 = 9 \times 4, since 9 boxes of 4 make 36.
7 and 9, because 7 \times 9 = 63.
(a) True, 3 \times 6 = 18. (b) True, 6 \times 3 = 18. (c) False, 2 \times 8 = 16, not 18.
24 more, since 32 - 8 = 24. 4 times as many, since 4 \times 8 = 32.
Chirag has 5 + 4 = 9 stickers. Killa has 4 \times 5 = 20 stickers.
42 more, since 48 - 6 = 42; 8 times as many, since 8 \times 6 = 48.
(a) 3 \times 6 = 18 stickers. (b) 6 + 3 = 9 stickers.
Killa’s pile is one box and Chirag’s is 3 boxes, so the 2 extra boxes make the 12. Each box is 12 \div 2 = 6. Killa has 6 stickers and Chirag has 3 \times 6 = 18. Check: 18 - 6 = 12.
No. 24 is 3 times as many as 8, but it is 24 - 8 = 16 more than 8. Three more than 8 is 11.
28 more, since 35 - 7 = 28; 5 times as many, since 5 \times 7 = 35.
240 meters. The bigger amount is missing, so multiply: 8 \times 30 is 8 \times 3 = 24 tens, which is 240.
63 stickers: 9 \times 7 = 63.
40 meters. The smaller amount is missing: 5 \times \square = 200, so \square = 200 \div 5 = 40, because 20 tens shared into 5 parts is 4 tens.
8 stickers: 7 \times \square = 56, so \square = 56 \div 7 = 8.
6 times as tall: \square \times 70 = 420, and 42 tens is 6 copies of 7 tens, so \square = 420 \div 70 = 6.
6 \times \square = 48, so \square = 48 \div 6 = 8 stickers on one sheet.
26 stickers. Killa has 3 \times 7 = 21, and Manaia has 21 + 5 = 26.
Lay out 30 items as 5 rows of 6. Each row is one copy of 6, so the array is 5 copies of 6: 30 is 5 times as many as 6. Read the same array by columns: there are 6 columns of 5, so 30 is also 6 copies of 5, which is 6 times as many as 5. No item moved, so both sentences describe the same 30.
No. 6 \times 4 = 24 is too small and 7 \times 4 = 28 is too big, so no whole number of copies of 4 makes 25. Six fours leave 1 over.
Draw the amount as one box and the bigger amount as 4 boxes of the same size. Line the one box up under the first of the four. That box has a partner, and the other 4 - 1 = 3 boxes stick out: they are the gap. So the gap is 3 copies. For 9: 4 \times 9 = 36, and the gap is 3 \times 9 = 27. Check: 36 - 9 = 27.